A functor \(K:\cal A\rightarrow \cal C\)  is dense if colimit preserving functors \(\cal C\rightarrow \cal B\)  are determined by their restriction along \(K\) , or, more formally, if the functor \([K,\mathsf {Id}_{\cal B}]:[\cal C, \cal B]\rightarrow [\cal A,\cal B]\)  restricting along \(K\)  is fully faithful [1]}. If, moreover, \(K\)  itself is fully faithful, then a colimit preserving functor \(\cal C\rightarrow \cal B\)  is the left Kan extension of its restriction along \(K\)  [1]}. Furthermore, we may be able to compute left Kan extensions explicitely with the help of a so-called density presentation [1]}. For example, we know (see [4]}) that \(\mathsf {D}:\mathbf {Set}\rightarrow \mathbf {Pos}\)  is dense and has a density presentation given by reflexive coinserters of `nerves of posets'. Explicitly, every poset \(X\)  is the reflexive coinserter
The coinserter of \(@C=8ex{VX^\mathbb {2}@<1pt>[r]^{}@<-1pt>[r]_{} & VX}\) 
in \(\mathbf {Set}\)  provides an example of weighted colimit that is not preserved by \(\mathsf {D}\) , showing that \(\mathsf {D}\)  cannot have an (enriched) right adjoint.
\(\hfill @C=8ex{\mathsf {D}\mathsf {V}X^\mathbb {2}@<2pt>[r]^{\mathsf {D}\pi _1}@<-2pt>[r]_{\mathsf {D}\pi _0} & \mathsf {D}\mathsf {V}X[r]^{}@/_1.5pc/[l]|-i & X}\hfill \) 
